The final element
INTRODUCTION No one knows how large an atomic nucleus can be. It is possible, though, to construct a boundary outside of which there is no realistic possibility of finding any nuclei. As a matter of definition, both nuclear drops and nuclides are evaluated without considering their environment. High pressure, in particular, can stabilize large blobs of nuclear matter (see "Neutron Star", this wiki); but the properties of such things are determined by local conditions. Nuclear drops and nuclides must be independent of their environment. In addition, nuclei must survive at zero pressure long enough for electromagnetic interactions to become important, around 10^-14 sec. Electromagnetic interactions are what allow nuclides to reach their ground state by emitting photons and what allow nuclei to bind electrons. Only two processes need to be considered when distinguishing nuclides from nuclear drops: neutron emission and spontaneous fission. NEUTRON EMISSION Neutron capture and emission are strong-force interactions, so they can happen in the time it takes a neutron to physically leave a nuclear potential well, on the order of 10^-23 sec. Neutrons within a drop of nuclear matter may be confined within the drop (bound) or simply happen to be within it (unbound). Since there is nothing to hinder their departure, unbound neutrons will leave a nuclear drop in less than 1E-14 (10^-14) sec. A nucleus can have only bound neutrons. There is a maximum number, Nd(Z), which Z protons can bind into a nuclear potential well. The locus of points (Z,Nd) form a curve called the neutron dripline. The problem is that no one knows exactly where the dripline is. It's an active area of theoretical research, but most study is focused on smaller nuclides (A <= 300). A few studies, though, do include predictions for dripline location which extend beyond A = 300. The KTUY model include a complete set of (Z,Nk) values for N≤310(1). Pages 15 & 18 of this reference show that even-N nucludes can be distinctly more neutron-rich than adjacent odd-N nuclides, so only even-N nuclides need be considered when definining an outer bound to what is possible. A suitable place to begin is at Nk = 66 = (50+82)/2. The pattern seen at N=126 and 184 - a nuclide at (shell closure +1) which is very neutron rich for an odd-N nuclide, followed by a neutron-rich nuclide at (closure+2) - extends to Nk = 310. Beyond that, the data become questionable. Manually extracted Z(Nk) data were converted Nk(Z) points and fitted with curves using Excel 2010. Exponential and logarithmic forms gave a poor fit, while power (p), quadratic (q) and linear (l) curves all gave good fits, with R^2 > 0.995 in all cases. These equations are: (K1) Nk,p(Z) = 2.3339*(Z^0.9988 (K2) Nk,q(Z) = 0.002399*Z2 + 1.9698*Z + 10.75 (K3) Nk,l(Z) = 2.3518*Z - 0.2596. Basu et al(2) also provide data for a table of (Z,Nm) values up to Z = 118. These data produced the equations: (B1) Nb,p(Z) = 1.7958*(Z^1.0717) (B2) Nb,q(Z\ (0.001525*(Z^2)) + 2.3897*Z - 4.379 (B3) Nb,l(Z) = 2.6139*Z - 11.59 R^2 also exceeds 0.995 for these equations. Note that the power law exponent is above 1 in Eq (B1), unlike Eq (K1), and that a2 is greater in Eq (K1) than Eq (B1). It is not obvious which set of equations extrapolates better to really large nuclei. A set of modeled beta-decay information for A < 340 was extracted in Fortran format from Reference 3. Given the nature of those data, the maximum value of N for each Z may be regarded as defining a neutron dripline. When curves were fitted to the (Z,Nm) values as was done in the previous cases, (M1) Nm,p(Z) = 3.158*(Z^0.9297) (M2) Nm,q(Z) = (0.000725*(Z^2)) + 2.1348*Z + 10.55 (M3) Nm,l(Z) = 2.2232*Z + 8.197 resulted. The data used did not display even-odd serrations in the way data in Ref 1 and 2 do. In addition, R^2 was high enough to raise the possibility that the values used represent a linear cutoff rather than a dripline. This set of equations is not as reliable as the first two, but does form a third input. One additional document(4) showed an extrapolated neutron dripline, but available document quality did not permit extraction of a fourth input set. Checking N(Z) values for large (Z > 200) makes clear that Eq. (K2) dominates at large Z. However, asimilar computation at Z=100 shows Eq (B2) returning the highest value. No one equation returns the maximum value for N(Z) over Z's entire range. Eq( K2)'s dominance at high Z means it is the necessary starting point for extrapolating the dripline to large Z. The ratio, rab(Z) = Na,b(Z) / Nk,q(Z) compares the value of N returned by the equations (K1) through (M3) with the corresponding value given by Eq. (K2). For each value of Z, there will be one equation for which rab(Z) is maximal; and there is also a maximum number, M, for the highest value of rab(Z) at any Z. Define the function N'(Z) by N'(Z) = M * Nk,q(Z). This function has the property that N'(Z) ≥ Na,b(Z) for all of Eq (K1) through (M3) at all values of Z. Figure 3 of Reference 4 includes a region called "benchmark" driplines. N'(Z) does not lie above extremes given in that document, but M*N'(Z) does - by a ratio of 1.03. Thus N''(Z) = M^2 * Nk,q(Z) lies above even "benchmark" nuclei. One final safety factor is warranted because nature always has the capacity to surprise. Since making the final Nd(Z) too big doesn't compromise its value as a boundary, squaring M^2 seems to be a choice which is cautious without being excessively timid. The computed value of M, 1.08052, yields. Nd(Z) = M^2 * N''(Z) = M^4 * Nk,q(Z) = 1.36311 * Nk,q(Z) so, from Eq (K2) (ND) Nd(Z) = 0.003270*(Z^2) + 2.69503*Z + 14.6533 gives a safe value for the maximum number of neutrons a nucleus with Z protons can have. FISSION Fission is the other factor limiting nuclear size. Actual stability against fission is strongly affected by shell closure(1),(6), but the underlying liquid-drop behavior can be described analytically. Reference 5 derives the potential energy barrier which a liquid-drop nucleus must overcome in order to fission. Since the derivation starts from the SEMF and does not include any explicit size restrictiocan be used for extrapolation to large nuclei., the equation : (F1a) fb{Z,A} = (98/1 ere, (F1b) x ≈ aC/(2aS) *(Z^2/A) can be used for extrapolation to large nuclei. Note that fb(Z,A) can be negative, which means a nuclear drop will come apart before it has completely formed. Ref. 6 provides a map of fission barriers. From this, and the global half-lives given on p. 15 of Ref. 1, it appears that half-life against fission will be around 1 sec. at 4 MeV barrier energy and 1E-09 sec at a barrier height to 2 MeV. This implies that a fission barrier on the order of 3 MeV is needed to stabilize a nuclear drop enough to permit beta decay, and a barrier on th fb(A,e order of 1 MeV is needed to prevent fission in less than 1E-14 sec. Define the term "needed structural correction energy, NCE(Z,A) by fb(Z,A) + NCE(Z,A) = X where X = 1 MeV where survival for 1E-14 sec is the issue and X = 3 MeV where the possibility of beta decay is the issue. Since X is known and fb(Z,A) can be computed for any Z and A, the expression NCE(Z,A) = X - fb(Z,A) can be plotted to give a map of the energy to stabilize any given nuclear drop against fission. Unlike the situation at the neutron dripline, where neutron emission goes goes from being the dominant decay mode to being unimportant with a change of even one neutron, fission half-lives increase more gradually. Since no abrupt change in decay mode is expected, the only way to define Nf(Z), the minimum number of neutrons needed for a nuclear drop to be a nucleus, is to determine NCE(Z,A) for extreme cases, then to define Nf(Z) as a set of points known to lie well beyond any of the extremes considered. NCE(138,264) and NCE(184,476) were considered, but the best extreme point seems to be NCE(173,440), which appears in Ref. 1. It is probably an artifact, but taking too large a value is not a problem when defining a boundary outside of which there are no nuclei. Using equations (F1a) and (F1b) gives fb(173,440) = -22.32 MeV. Although X = 1 MeV is a more realistic value for distinguishing drops which fission in less than 1E-14 sec, use of X = 3 MeV adds a further measure of caution giving NCE(173,440) = 3- (-22.32) = 25.32 MeV. While it isn't likely any possible structural correction can contribute more than 25.32 MeV toward stabilizing a nuclide, using NEC(Z,A) = 2*25.32 = 50.64 MeV provides strong assurance that stronger corrections are essentially impossible. No analytic expression for an Nf(Z) curve obtained by holding NEC(Z,(Z+Nf)) constant, it is possible to plot NEC(Z,A) and to extract an Nf(Z) curve that way. Such a curve serves as a reliable boundary between nuclear drops which might fission and those which must fission in a time frame below 1E-14 sec. OUTER BOUNDARY Must-fission curves rise more rapidly with Z than does Nd(Z), so there will be a maximum Z for which Nf(Z) <= Nd(Z) Using equation (ND) and a must-fission curve tied to NEC(Z,A) = 50.64 produces convergence of the two curves at Z = 324. The chosen dripline and must-fission curves form a boundary between nuclear drops which might be nuclides and those which cannot be. Since all choices made in generating those curves were biased toward maximizing the region within which nuclides are possible, those two curves create a firm outer boundary to what is possible. INNER BOUNDARY While the outer boundary defined above is reliable, it will include a great many nuclear drops whose prospects for being nuclides are poor. A second, inner boundary within which nuclides are expected rather than merely possible, is needed to more realistically bound the region in which smaller nuclides occur. Eq. (K2) can be used to describe a neutron dripline outside of which nuclei rarely occur (but are not impossible). Since the largest NCE for any drop realistically predicted to be a nuclide in Ref 1 is NCE(175,471) = 17.40, using twice that value to define a must-fission curve is both conservative and realistic. REFERENCES # "Decay Modes and a Limit of Existence of Nuclei"; H. Koura; The 4th International Conference on the Chemistry and Physics of the Transactinide Elements; url: http:// tan11.jinr.ru /pdf/10_Sep/S_2/05_ Koura.pdf. # "Neutron and Proton Drip Lines Using the Modified Bethe-Weizsacker Mass Formula; D.N. Basu et al; Int.J.Mod.Phys.; arXiv:nucl-th/0306061; url: https://arxiv.org/abs/nucl-th/0306061 3. "Nuclear Astrophysics Data" / "Nuclear Properties for Astrophysical Applications"; P. Möller, et al url: https://t2.lanl.gov/nis/data/astro/molnix96/molnix.html 4. "Positioning the Neutron Drip Line and the R Process Paths in the Nuclear Landscape"; R. Wang and L.W. Chen; Journal reference: Phys. Rev. C 92, 031303® (2015); DOI \10.1103/ PhysRevC. 92.031303; arXiv:1410.2498 nucl-th 5. "Semi Empirical: Liquid Drop Model and Nuclear Fission"; Crawford; (course notes extract); url: http://inpp.ohiou.edu/~crawford/phys7501-liquidDrop.pdf 6. "Fission Mechanism of Exotic Nuclei"; Soshiki et al; Research Group for Heavy Element Nuclear Science (no further details); url: http://asrc.jaea.go.jp/soshiki/gr/HENS-gr/np/research/ pageFission_e.html 7. "Magic Numbers of Ultraheavy Nuclei"; Denisov, V.; Physics of Atomic Nuclei. 68 (7): 1133–1137. doi:10.1134/1.1992567. 8. "Single-Particle Levels of Spherical Nuclei in the Superheavy and Extremely Superheavy Mass Region"; Hiroyuki Koura & Satoshi Chiba; Journal of the Physical Society of Japan. 82: 014201. doi:10.7566/JPSJ.82.014201. (12-23-19)Category:Undiscovered elements Category:Element article stubs Category:Elements Category:Radioactive